Properties

Label 2004.815
Modulus $2004$
Conductor $2004$
Order $166$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2004)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([83,83,164]))
 
pari: [g,chi] = znchar(Mod(815,2004))
 

Basic properties

Modulus: \(2004\)
Conductor: \(2004\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(166\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2004.o

\(\chi_{2004}(11,\cdot)\) \(\chi_{2004}(47,\cdot)\) \(\chi_{2004}(107,\cdot)\) \(\chi_{2004}(179,\cdot)\) \(\chi_{2004}(191,\cdot)\) \(\chi_{2004}(203,\cdot)\) \(\chi_{2004}(215,\cdot)\) \(\chi_{2004}(239,\cdot)\) \(\chi_{2004}(251,\cdot)\) \(\chi_{2004}(263,\cdot)\) \(\chi_{2004}(275,\cdot)\) \(\chi_{2004}(299,\cdot)\) \(\chi_{2004}(311,\cdot)\) \(\chi_{2004}(359,\cdot)\) \(\chi_{2004}(383,\cdot)\) \(\chi_{2004}(395,\cdot)\) \(\chi_{2004}(419,\cdot)\) \(\chi_{2004}(431,\cdot)\) \(\chi_{2004}(455,\cdot)\) \(\chi_{2004}(467,\cdot)\) \(\chi_{2004}(491,\cdot)\) \(\chi_{2004}(503,\cdot)\) \(\chi_{2004}(515,\cdot)\) \(\chi_{2004}(539,\cdot)\) \(\chi_{2004}(551,\cdot)\) \(\chi_{2004}(563,\cdot)\) \(\chi_{2004}(599,\cdot)\) \(\chi_{2004}(623,\cdot)\) \(\chi_{2004}(671,\cdot)\) \(\chi_{2004}(695,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((1003,1337,673)\) → \((-1,-1,e\left(\frac{82}{83}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{81}{166}\right)\)\(e\left(\frac{13}{166}\right)\)\(e\left(\frac{55}{83}\right)\)\(e\left(\frac{63}{83}\right)\)\(e\left(\frac{143}{166}\right)\)\(e\left(\frac{133}{166}\right)\)\(e\left(\frac{67}{83}\right)\)\(e\left(\frac{81}{83}\right)\)\(e\left(\frac{115}{166}\right)\)\(e\left(\frac{69}{166}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{83})$
Fixed field: Number field defined by a degree 166 polynomial